Question

The set $ S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\} $ is to be partitioned into three sets $ A, B, C $ of equal size. Thus $ A \cup B \cup C = S $, $ A \cap B = B \cap C = A \cap C = \emptyset $. The number of ways to partition $ S $ is

• A. \( \frac{12!}{(4!)^3} \)
• B. \( \frac{12!}{3! (4!)^3} \)
• C. \( \frac{12!}{(4!)^4} \)
• D. \( \frac{12!}{3! (4!)^4} \)

Hint:

The formula for partitioning a set of \( n \) elements into subsets of equal size \( k \) is \( \frac{n!}{(k!)^m \times m!} \), where \( m \) is the number of subsets.

Answer 1

The Correct Option is A

We need to partition a set of 12 elements into three subsets \( A \), \( B \), and \( C \), each containing 4 elements. The total number of ways to do this is given by: \[ \frac{12!}{(4!)^3 \times 3!} \] However, the correct expression for the number of ways to partition the set is already simplified in the options as: \[ \frac{12!}{(4!)^3} \]
Therefore, the correct answer is 1. \( \frac{12!}{(4!)^3} \).